Methods and Systems for Conserving Power During Hover Flight

ABSTRACT

An example method may include determining a drag force of an apparent wind on an aircraft that is coupled to a ground station via a tether. The method also includes determining a trajectory of the aircraft to a point downwind of the ground station such that the aircraft travelling the trajectory causes the tether to unfurl along a catenary path above ground. The method further includes determining an orientation of the aircraft to travel the trajectory in the apparent wind so that an actuator of the aircraft is configured to provide a vertical thrust in a direction substantially perpendicular to the ground. The method also includes determining a vertical thrust for the aircraft at the orientation to travel the trajectory in the apparent wind. The method also includes providing instructions to cause the actuator of the aircraft to provide the vertical thrust to move the aircraft along the trajectory.

BACKGROUND

Unless otherwise indicated herein, the materials described in thissection are not prior art to the claims in this application and are notadmitted to be prior art by inclusion in this section.

Many techniques and systems exist for controlling a flight path of anaircraft. Generally, an ability to change a position or an attitude ofthe aircraft will depend on the location and functionality of actuatorsincluded as part of the aircraft.

SUMMARY

In one example, a method is provided that includes determining a dragforce of an apparent wind on an aircraft coupled to a ground station viaa tether. The method also includes, based on the drag force and a weightof the tether, determining a trajectory of the aircraft to a pointdownwind of the ground station such that the aircraft travelling thetrajectory causes the tether to unfurl along a catenary path aboveground. The method further includes determining an orientation of theaircraft to travel the trajectory in the apparent wind so that anactuator of the aircraft is configured to provide a vertical thrust in adirection substantially perpendicular to the ground. The method alsoincludes, based on the trajectory and a weight of the aircraft,determining a vertical thrust for the aircraft at the orientation totravel the trajectory in the apparent wind. The method also includesproviding instructions to cause the actuator of the aircraft to providethe vertical thrust to move the aircraft along the trajectory.

In another example, a computer readable storage memory having storedtherein instructions, that when executed by a computing device thatincludes one or more processors, cause the computing device to performfunctions is provided. The functions comprise determining a drag forceof an apparent wind on an aircraft coupled to a ground station via atether. The functions further comprise, based on the drag force and aweight of the tether, determining a trajectory of the aircraft to apoint downwind of the ground station such that the aircraft travellingthe trajectory causes the tether to unfurl along a catenary path aboveground. The functions further comprise determining an orientation of theaircraft to travel the trajectory in the apparent wind so that anactuator of the aircraft is configured to provide a vertical thrust in adirection substantially perpendicular to the ground. The functionsfurther comprise based on the trajectory and a weight of the aircraft,determining a vertical thrust for the aircraft at the orientation totravel the trajectory in the apparent wind. The functions furthercomprise providing instructions to cause the actuator of the aircraft toprovide the vertical thrust to move the aircraft along the trajectory.

In still another example, a system is provided that comprises one ormore processors and memory configured to store instructions, that whenexecuted by the one or more processors, cause the system to performfunctions. The functions comprise determining a drag force of anapparent wind on an aircraft coupled to a ground station via a tether.The functions further comprise, based on the drag force and a weight ofthe tether, determining a trajectory of the aircraft to a point downwindof the ground station such that the aircraft travelling the trajectorycauses the tether to unfurl along a catenary path above ground. Thefunctions further comprise determining an orientation of the aircraft totravel the trajectory in the apparent wind so that an actuator of theaircraft is configured to provide a vertical thrust in a directionsubstantially perpendicular to the ground. The functions furthercomprise, based on the trajectory and a weight of the aircraft,determining a vertical thrust for the aircraft at the orientation totravel the trajectory in the apparent wind. The functions furthercomprise providing instructions to cause the actuator of the aircraft toprovide the vertical thrust to move the aircraft along the trajectory.

In yet another example, a system is provided that includes a means fordetermining a drag force of an apparent wind on an aircraft coupled to aground station via a tether. The system further comprises means for,based on the drag force and a weight of the tether, determining atrajectory of the aircraft to a point downwind of the ground stationsuch that the aircraft travelling the trajectory causes the tether tounfurl along a catenary path above ground. The system further comprisesmeans for determining an orientation of the aircraft to travel thetrajectory in the apparent wind so that an actuator of the aircraft isconfigured to provide a vertical thrust in a direction substantiallyperpendicular to the ground. The system further comprises means for,based on the trajectory and a weight of the aircraft, determining avertical thrust for the aircraft at the orientation to travel thetrajectory in the apparent wind. The system further comprises means forproviding instructions to cause the actuator of the aircraft to providethe vertical thrust to move the aircraft along the trajectory.

These as well as other aspects, advantages, and alternatives, willbecome apparent to those of ordinary skill in the art by reading thefollowing detailed description, with reference where appropriate to theaccompanying figures.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts a tethered flight system, according to an exampleembodiment.

FIG. 2 is a simplified block diagram illustrating example components ofthe tethered flight system.

FIG. 3A depicts a downward looking view of an example tethered flightsystem.

FIG. 3B depicts examples of the aircraft engaging in hover flight atvarious horizontal positions and altitudes.

FIG. 4A depicts a first example catenary path and a second examplecatenary path.

FIG. 4B depicts a third example catenary path and a fourth examplecatenary path.

FIG. 5A depicts an example roll axis of an aircraft.

FIG. 5B depicts an example pitch axis of the aircraft.

FIG. 5C depicts an example yaw axis of the aircraft.

FIG. 6A depicts examples of a pitch axis of an aircraft, a tail wing,and an apparent wind.

FIG. 6B depicts examples of a pitch axis of an aircraft, a tail wing,and an apparent wind.

FIG. 7 is a block diagram of an example method for determining atrajectory and an orientation of the aircraft that causes a tether tounfurl along a catenary path above ground.

DETAILED DESCRIPTION

The following detailed description describes various features andfunctions of the disclosed systems and methods with reference to theaccompanying figures. In the figures, similar symbols identify similarcomponents, unless context dictates otherwise. The illustrative systemand method embodiments described herein are not meant to be limiting. Itmay be readily understood that certain aspects of the disclosed systemsand methods can be arranged and combined in a wide variety of differentconfigurations, all of which are contemplated herein.

Within examples, a processor may be configured to determine a drag forceof an apparent wind on an aircraft tethered to a ground station. Theprocessor may determine the drag force based on a density of air, a dragcoefficient of the aircraft, a reference area of the aircraft, or aspeed of the apparent wind. The drag coefficient may represent atendency of the aircraft to resist movement of air moving over thesurface of the aircraft based on the shape of the aircraft. Thereference area of the aircraft may represent a cross sectional area ofthe aircraft in a plane perpendicular to the apparent wind, but may alsorepresent any area of the aircraft.

Next, the processor may determine a trajectory of the aircraft to apoint downwind of the ground station such that by moving along thetrajectory, the aircraft pulls the tether along a catenary path aboveground as the tether is unfurled. The processor may determine thetrajectory based on the drag force and a weight of the tether, such thata tension of the tether is caused by the drag force of the apparentwind. For example, a decrease in the weight of the tether or an increasein the drag force may cause the point downwind of the ground station tobe at a lower altitude. By further example, an increase in the weight ofthe tether or a decrease in the drag force may cause the point downwindof the ground station to be at a higher altitude.

The processor may also determine an orientation of the aircraft for theaircraft to travel toward the point downwind of the ground station. Theaircraft may include an actuator and while the aircraft is in theorientation the actuator may be configured to provide a vertical thrustin a direction substantially perpendicular to the ground. Theorientation may be referred to as zero pitch. The aircraft being in theorientation may allow the actuator to move the aircraft in asubstantially vertical direction, while the drag force from the apparentwind moves the aircraft in a substantially horizontal direction.

Based on the trajectory and a weight of the aircraft, the processor maydetermine a vertical thrust for the aircraft at the orientation totravel the trajectory in the apparent wind. The processor may determinea vertical acceleration to travel the trajectory, and may determine thevertical thrust based on the vertical acceleration, the weight of theaircraft, a weight of a portion of the tether supported by the aircraft,and gravitational forces acting on the aircraft and the tether. Theprocessor may further provide instructions to the actuator to providethe vertical thrust to move the aircraft along the trajectory.

Referring now to the figures, FIG. 1 depicts a tethered flight system100, according to an example embodiment. The tethered flight system 100may include a ground station 110, a tether 120, and an aircraft 130. Asshown in FIG. 1, the aircraft 130 may be connected to the tether 120,and the tether 120 may be connected to the ground station 110. Thetether 120 may be attached to the ground station 110 at one location onthe ground station 110, and attached to the aircraft 130 at twolocations on the aircraft 130. However, in other examples, the tether120 may be attached at multiple locations to any part of the groundstation 110 or the aircraft 130.

The ground station 110 may be used to hold or support the aircraft 130until the aircraft 130 is in a flight mode. The ground station 110 mayalso be configured to reposition the aircraft 130 such that deployingthe aircraft 130 is possible. Further, the ground station 110 may befurther configured to receive the aircraft 130 during a landing. Theground station 110 may be formed of any material that can suitably keepthe aircraft 130 attached or anchored to the ground while in hoverflight, forward flight, or crosswind flight.

In addition, the ground station 110 may include one or more components(not shown), such as a winch, that may vary a length of the tether 120.For example, when the aircraft 130 is deployed, the one or morecomponents may be configured to pay out or reel out the tether 120. Insome implementations, the one or more components may be configured topay out or reel out the tether 120 to a predetermined length. Asexamples, the predetermined length could be equal to or less than amaximum length of the tether 120. Further, when the aircraft 130 landson the ground station 110, the one or more components may be configuredto reel in the tether 120.

The tether 120 may transmit electrical energy generated by the aircraft130 to the ground station 110. In addition, the tether 120 may transmitelectricity to the aircraft 130 to power the aircraft 130 for takeoff,landing, hover flight, or forward flight. The tether 120 may beconstructed in any form and using any material which allows for thetransmission, delivery, or harnessing of electrical energy generated bythe aircraft 130 or transmission of electricity to the aircraft 130. Thetether 120 may also be configured to withstand one or more forces of theaircraft 130 when the aircraft 130 is in a flight mode. For example, thetether 120 may include a core configured to withstand one or more forcesof the aircraft 130 when the aircraft 130 is in hover flight, forwardflight, or crosswind flight. The core may be constructed of highstrength fibers. In some examples, the tether 120 may have a fixedlength or a variable length.

The aircraft 130 may include various types of devices, such as a kite, ahelicopter, a wing, or an airplane, among other possibilities. Theaircraft 130 may be formed of solid structures of metal, plastic,polymers, or any material which allows for a high thrust-to-weight ratioand generation of electrical energy which may be used in utilityapplications. Additionally, the materials may allow for a lightninghardened, redundant or fault tolerant design which may be capable ofhandling large or sudden shifts in wind speed and wind direction. Othermaterials may be possible as well.

As shown in FIG. 1, the aircraft 130 may include a main wing 131, afront section 132, actuator connectors 133A-B, actuators 134A-D, a tailboom 135, a tail wing 136, and a vertical stabilizer 137. Any of thesecomponents may be shaped in any form which allows for the use of lift toresist gravity or move the aircraft 130 forward.

The main wing 131 may provide a primary lift for the aircraft 130 duringforward flight, wherein the aircraft 130 may move through air in adirection substantially parallel to a direction of thrust provided bythe actuators 134A-D so that the main wing 131 provides a lift forcesubstantially perpendicular to a ground. The main wing 131 may be one ormore rigid or flexible airfoils, and may include various controlsurfaces or actuators, such as winglets, flaps, rudders, elevators, etc.The control surfaces may be used to steer or stabilize the aircraft 130or reduce drag on the aircraft 130 during hover flight, forward flight,or crosswind flight. The main wing 131 may be any suitable material forthe aircraft 130 to engage in hover flight, forward flight, or crosswindflight. For example, the main wing 131 may include carbon fiber ore-glass. Moreover, the main wing 131 may have a variety dimensions. Forexample, the main wing 131 may have one or more dimensions thatcorrespond with a conventional wind turbine blade. The front section 132may include one or more components, such as a nose, to reduce drag onthe aircraft 130 during flight.

The actuator connectors 133A-B may connect the actuators 134A-D to themain wing 131. In some examples, the actuator connectors 133A-B may takethe form of or be similar in form to one or more pylons. In the exampledepicted in FIG. 1, the actuator connectors 133A-B are arranged suchthat the actuators 134A and 134B are located on opposite sides of themain wing 131 and actuators 134C and 134D are also located on oppositesides of the main wing 131. The actuator 134C may also be located on anend of the main wing 131 opposite of the actuator 134A, and the actuator134D may be located on an end of main wing 131 opposite of the actuator134B.

In a power generating mode, the actuators 134A-D may be configured todrive one or more generators for the purpose of generating electricalenergy. As shown in FIG. 1, the actuators 134A-D may each include one ormore blades. The actuator blades may rotate via interactions with thewind and could be used to drive the one or more generators. In addition,the actuators 134A-D may also be configured to provide a thrust to theaircraft 130 during flight. As shown in FIG. 1, the actuators 134A-D mayfunction as one or more propulsion units, such as a propeller. Althoughthe actuators 134A-D are depicted as four actuators in FIG. 1, in otherexamples the aircraft 130 may include any number of actuators.

In a forward flight mode, the actuators 134A-D may be configured togenerate a forward thrust substantially parallel to the tail boom 135.Based on the position of the actuators 134A-D relative to the main wing131 depicted in FIG. 1, the actuators may be configured to provide amaximum forward thrust for the aircraft 130 when all of the actuators134A-D are operating at full power. The actuators 134A-D may provideequal or about equal amounts of forward thrusts when the actuators134A-D are operating at full power, and a net rotational force appliedto the aircraft by the actuators 134A-D may be zero.

The tail boom 135 may connect the main wing 131 to the tail wing 136 andthe vertical stabilizer 137. The tail boom 135 may have a variety ofdimensions. Moreover, in some implementations, the tail boom 135 couldtake the form of a body or fuselage of the aircraft 130. In suchimplementations, the tail boom 135 may carry a payload.

The tail wing 136 or the vertical stabilizer 137 may be used to steer orstabilize the aircraft 130 or reduce drag on the aircraft 130 duringhover flight, forward flight, or crosswind flight. For example, the tailwing 136 or the vertical stabilizer 137 may be used to maintain a pitchor a yaw attitude of the aircraft 130 during hover flight, forwardflight, or crosswind flight. In FIG. 1, the vertical stabilizer 137 isattached to the tail boom 135, and the tail wing 136 is located on topof the vertical stabilizer 137. The tail wing 136 may have a variety ofdimensions.

While the aircraft 130 has been described above, it should be understoodthat the methods and systems described herein could involve any aircraftthat is connected to a tether, such as the tether 120.

FIG. 2 is a simplified block diagram illustrating example components ofthe tethered flight system 200. The tethered flight system 200 mayinclude the ground station 210, the tether 220, and the aircraft 230. Asshown in FIG. 2, the ground station 210 may include one or moreprocessors 212, data storage 214, program instructions 216, and acommunication system 218. A processor 212 may be a general-purposeprocessor or a special purpose processor (e.g., digital signalprocessors, application specific integrated circuits, etc.). The one ormore processors 212 may be configured to execute computer-readableprogram instructions 216 that are stored in data storage 214 and areexecutable to provide at least part of the functionality describedherein.

The data storage 214 may include or take the form of one or morecomputer-readable storage media that may be read or accessed by at leastone processor 212. The one or more computer-readable storage media caninclude volatile or non-volatile storage components, such as optical,magnetic, organic or other memory or disc storage, which may beintegrated in whole or in part with at least one of the one or moreprocessors 212. In some embodiments, the data storage 214 may beimplemented using a single physical device (e.g., one optical, magnetic,organic or other memory or disc storage unit), while in otherembodiments, the data storage 214 can be implemented using two or morephysical devices.

As noted, the data storage 214 may include computer-readable programinstructions 216 and perhaps additional data, such as diagnostic data ofthe ground station 210. As such, the data storage 214 may includeprogram instructions to perform or facilitate some or all of thefunctionality described herein.

In a further respect, the ground station 210 may include thecommunication system 218. The communications system 218 may include oneor more wireless interfaces or one or more wireline interfaces, whichallow the ground station 210 to communicate via one or more networks.Such wireless interfaces may provide for communication under one or morewireless communication protocols, such as Bluetooth, WiFi (e.g., an IEEE802.11 protocol), Long-Term Evolution (LTE), WiMAX (e.g., an IEEE 802.16standard), a radio-frequency ID (RFID) protocol, near-fieldcommunication (NFC), or other wireless communication protocols. Suchwireline interfaces may include an Ethernet interface, a UniversalSerial Bus (USB) interface, or a similar interface to communicate via awire, a twisted pair of wires, a coaxial cable, an optical link, afiber-optic link, or other physical connection to a wireline network.The ground station 210 may communicate with the aircraft 230, otherground stations, or other entities (e.g., a command center) via thecommunication system 218.

In an example embodiment, the ground station 210 may includecommunication systems 218 that allows for both short-range communicationand long-range communication. For example, the ground station 210 may beconfigured for short-range communications using Bluetooth and forlong-range communications under a CDMA protocol. In such an embodiment,the ground station 210 may be configured to function as a “hot spot”, oras a gateway or proxy between a remote support device (e.g., the tether220, the aircraft 230, and other ground stations) and one or more datanetworks, such as a cellular network or the Internet. Configured assuch, the ground station 210 may facilitate data communications that theremote support device would otherwise be unable to perform by itself.

For example, the ground station 210 may provide a WiFi connection to theremote device, and serve as a proxy or gateway to a cellular serviceprovider's data network, which the ground station 210 might connect tounder an LTE or a 3G protocol, for instance. The ground station 210could also serve as a proxy or gateway to other ground stations or acommand station, which the remote device might not be able to otherwiseaccess.

Moreover, as shown in FIG. 2, the tether 220 may include transmissioncomponents 222 and a communication link 224. The transmission components222 may be configured to transmit electrical energy from the aircraft230 to the ground station 210 or transmit electrical energy from theground station 210 to the aircraft 230. The transmission components 222may take various different forms in different embodiments. For example,the transmission components 222 may include one or more conductors thatare configured to transmit electricity. And in at least one suchexample, the one or more conductors may include aluminum or any othermaterial which allows for the conduction of electric current. Moreover,in some implementations, the transmission components 222 may surround acore of the tether 220 (not shown).

The ground station 210 could communicate with the aircraft 230 via thecommunication link 224. The communication link 224 may be bidirectionaland may include one or more wired or wireless interfaces. Also, therecould be one or more routers, switches, or other devices or networksmaking up at least a part of the communication link 224.

Further, as shown in FIG. 2, the aircraft 230 may include one or moresensors 232, a power system 234, power generation/conversion components236, a communication system 238, one or more processors 242, datastorage 244, program instructions 246, and a control system 248.

The sensors 232 could include various different sensors in differentembodiments. For example, the sensors 232 may include a globalpositioning system (GPS) receiver. The GPS receiver may be configured toprovide data that is typical of GPS systems (which may be referred to asa global navigation satellite system (GNNS)), such as the GPScoordinates of the aircraft 230. Such GPS data may be utilized by thetethered flight system 200 to provide various functions describedherein.

As another example, the sensors 232 may include one or more windsensors, such as one or more pitot tubes. The one or more wind sensorsmay be configured to detect apparent or relative wind. Such wind datamay be utilized by the tethered flight system 200 to provide variousfunctions described herein.

Still as another example, the sensors 232 may include an inertialmeasurement unit (IMU). The IMU may include both an accelerometer and agyroscope, which may be used together to determine the orientation orattitude of the aircraft 230. In particular, the accelerometer canmeasure the orientation of the aircraft 230 with respect to earth, whilethe gyroscope measures the rate of rotation around an axis, such as acenterline of the aircraft 230. IMUs are commercially available inlow-cost, low-power packages. For instance, the IMU may take the form ofor include a miniaturized MicroElectroMechanical System (MEMS) or aNanoElectroMechanical System (NEMS). Other types of IMUs may also beutilized. The IMU may include other sensors, in addition toaccelerometers and gyroscopes, which may help to better determineposition. Two examples of such sensors are magnetometers and pressuresensors. Other examples are also possible.

While an accelerometer and gyroscope may be effective at determining theorientation of the aircraft 230, errors in measurement may compound overtime. However, an example aircraft 230 may be able mitigate or reducesuch errors by using a magnetometer to measure direction. One example ofa magnetometer is a low-power, digital 3-axis magnetometer, which may beused to realize an orientation independent electronic compass foraccurate heading information. However, other types of magnetometers maybe utilized as well.

The aircraft 230 may also include a pressure sensor or barometer, whichcan be used to determine the altitude of the aircraft 230.Alternatively, other sensors, such as sonic altimeters or radaraltimeters, can be used to provide an indication of altitude, which mayhelp to improve the accuracy of or prevent drift of the IMU. Theaircraft 230 may include a thermometer or another sensor that senses airtemperature as well.

As noted, the aircraft 230 may include the power system 234. The powersystem 234 could take various different forms in different embodiments.For example, the power system 234 may include one or more batteries thatprovide power to the aircraft 230. In some implementations, the one ormore batteries may be rechargeable and each battery may be recharged viaa wired connection between the battery and a power supply or via awireless charging system, such as an inductive charging system thatapplies an external time-varying magnetic field to an internal batteryor a charging system that uses energy collected from one or more solarpanels.

As another example, the power system 234 may include one or more motorsor engines for providing power to the aircraft 230. In one embodiment,the power system 234 may provide power to the actuators 134A-D of theaircraft 130, as shown and described in FIG. 1. In some implementations,the one or more motors or engines may be powered by a fuel, such as ahydrocarbon-based fuel. In such implementations, the fuel could bestored on the aircraft 230 and delivered to the one or more motors orengines via one or more fluid conduits, such as piping. In someimplementations, the power system 234 may be implemented in whole or inpart on the ground station 210.

As noted, the aircraft 230 may include the power generation/conversioncomponents 236. The power generation/conversion components 236 couldtake various different forms in different embodiments. For example, thepower generation/conversion components 236 may include one or moregenerators, such as high-speed, direct-drive generators. The one or moregenerators may be driven by one or more rotors or actuators, such as theactuators 134A-D as shown and described in FIG. 1.

Moreover, the aircraft 230 may include a communication system 238. Thecommunication system 238 may take the form of or be similar in form tothe communication system 218 of the ground station 210. The aircraft 230may communicate with the ground station 210, other aircrafts, or otherentities (e.g., a command center) via the communication system 238.

In some implementations, the aircraft 230 may be configured to functionas a “hot spot” or as a gateway or proxy between a remote support device(e.g., the ground station 210, the tether 220, other aircrafts) and oneor more data networks, such as cellular network or the Internet.Configured as such, the aircraft 230 may facilitate data communicationsthat the remote support device would otherwise be unable to perform byitself.

For example, the aircraft 230 may provide a WiFi connection to theremote device, and serve as a proxy or gateway to a cellular serviceprovider's data network, which the aircraft 230 might connect to underan LTE or a 3G protocol, for instance. The aircraft 230 could also serveas a proxy or gateway to other aircrafts or a command station, which theremote device might not be able to otherwise access.

As noted, the aircraft 230 may include the one or more processors 242,the program instructions 244, and the data storage 246. The one or moreprocessors 242 can be configured to execute computer-readable programinstructions 246 that are stored in the data storage 244 and areexecutable to provide at least part of the functionality describedherein. The one or more processors 242 may take the form of or besimilar in form to the one or more processors 212, the data storage 244may take the form of or be similar in form to the data storage 214, andthe program instructions 246 may take the form of or be similar in formto the program instructions 216.

Moreover, as noted, the aircraft 230 may include the control system 248.In some implementations, the control system 248 may be configured toperform one or more functions described herein. The control system 248may be implemented with mechanical systems or with hardware, firmware,or software. As one example, the control system 248 may take the form ofprogram instructions stored on a non-transitory computer readable mediumand a processor that executes the instructions. The control system 248may be implemented in whole or in part on the aircraft 230 or at leastone entity remotely located from the aircraft 230, such as the groundstation 210. Generally, the manner in which the control system 248 isimplemented may vary, depending upon the particular embodiment.

FIG. 3A depicts a downward looking view of an example tethered flightsystem 300 which may include a ground station 310, a tether 320, and anaircraft 330. Also depicted in FIG. 3A are an azimuth angle 340 and anapparent wind 350. As shown in FIG. 3A, the ground station 310 may becoupled to the tether 320 at a first end of the tether 320 while thetether 320 may be coupled to the aircraft 330 at a second end of thetether 320. The aircraft 330 may be configured to freely fly in anazimuthal direction about the ground station 310. A position of theaircraft 330 may be characterized in part by the azimuth angle 340between a reference angle and the azimuthal position of the aircraft330. The ground station 310 may be rotated so as to deploy the aircraft330 in a direction parallel to the apparent wind 350.

FIG. 3B depicts examples of the aircraft 330 engaging in hover flight atvarious horizontal positions and altitudes. The aircraft 330 may betethered to the ground station 310 via the tether 320. FIG. 3B alsodepicts the apparent wind 350, a ground 360, a horizontal distance 370,and an altitude 380 of the aircraft.

Hover flight may be characterized by the aircraft 330 travelling at anattitude such that a primary force resisting a force of gravity on theaircraft 330 is provided by the thrust of the actuators of the aircraft330. The aircraft 330 may be deployed in a direction parallel to theapparent wind 350. In such a configuration, the actuators may beoriented to provide thrust in a direction substantially perpendicular tothe ground 360 and the main wing may be oriented so that the main wingis not configured to apply a lift force to the aircraft 330 in adirection perpendicular to the ground 360. During hover flight, liftgenerating surfaces of the main wing, the tail wing, and the horizontalstabilizer may not be effective in generating lift as the liftgenerating surfaces may either be oriented to face substantiallyparallel to a direction of travel of the aircraft 330, or may not beimpacted with a sufficient apparent wind 350 to generate a lift force.In hover flight, forces causing the aircraft 330 to move along a flightpath may include forces provided by the actuators and the apparent wind350.

Hover flight may begin with deploying the aircraft 330 from the groundstation 310 in a hover-flight orientation. The ground station 310 may berotated so as to deploy the aircraft 330 in an azimuthal directionparallel with the apparent wind 350. Deploying the aircraft 330 in thedirection of the apparent wind 350 may enable the aircraft 330 to travelthe horizontal distance 370 from the ground station 310 while theactuators of aircraft 330 are thrusting in a substantially verticaldirection. The tether 320 may be paid out or reeled out as the aircraft330 achieves increasing horizontal distance 370 from the ground station310. Hover flight may include the aircraft 330 ascending, descending, orhovering over the ground 360 at an altitude 380 above the ground 360.

FIG. 4A depicts a first example catenary path 402 and a second examplecatenary path 404. A cable, chain, tether or similar object may hangalong a catenary path when the object is supported at a first end andsecond end, but otherwise allowed to freely hang and react togravitational forces.

Within examples, a catenary path of the tether can be equivalent (orabout equal) to the trajectory of the aircraft if a common point of thetether and the aircraft, i.e. a point where the tether connects to theaircraft, is defined to travel both the trajectory of the aircraft andthe catenary path of the tether. To calculate the catenary path of thetether, the processor may determine the drag force (F) on the aircraftdue to an apparent wind using an equation [1]:

$\begin{matrix}{F = {\frac{\rho}{2}C_{d}{Av}^{2}}} & \lbrack 1\rbrack\end{matrix}$

The determination may first include determining or receiving parametersof the equation [1] such as a density of air surrounding the aircraft(ρ), a drag coefficient of the aircraft (C_(d)), a reference area of theaircraft (A), and a speed of the apparent wind impacting the aircraft(v). The drag coefficient may be dependent on a number of variables,such as a shape of the aircraft and the speed of the apparent wind. Thereference area of the aircraft may be a cross-sectional area of theaircraft in a plane perpendicular to a direction of the apparent wind.However, the reference area may be any area of the aircraft. The dragforce on the aircraft may be proportional to the density of air, thedrag coefficient, the reference area, and a square of the speed of theapparent wind, as depicted in the equation [1]. Data representing theparameters may be received by the processor from sensors of the aircraftor ground station, or may be stored in memory. For example, theprocessor may determine the density of air based on receiving datarepresenting an air temperature and pressure, may receive datarepresenting the speed of the apparent wind, but may retrieve datarepresenting the drag coefficient and the reference area from memory.Once the drag force is determined, the drag force may be used as aparameter of an equation [2]:

$\begin{matrix}{h = {\frac{T_{0}}{\mu}\lbrack {{\cosh ( \frac{\mu \; x}{T_{0}} )} - 1} \rbrack}} & \lbrack 2\rbrack\end{matrix}$

The equation [2] may define a relationship between an altitude of thetether (h), the horizontal position of the tether (x), a tension of thetether (T₀) at a lowest point of the catenary path, and a length and aweight of the tether (or a weight per length of the tether (μ)). (T₀ mayalso represent a horizontal component of the tension at any point on thetether.) A “cos h” function may be a hyperbolic cosine function, whichmay be equivalently expressed as exponential functions as in an equation[3]:

$\begin{matrix}{h = {\frac{T_{0}}{\mu}\lfloor {( \frac{( {^{(\frac{\mu \; x}{T_{0}})} + ^{- {(\frac{\mu \; x}{T_{0}})}}} )}{2} ) - 1} \rfloor}} & \lbrack 3\rbrack\end{matrix}$

The equation [3] may define a relationship between an altitude of thetether (h), the horizontal position of the tether (x), a tension of thetether (T₀), and a length and a weight of the tether (or a weight perlength of the tether (μ)).

The drag force (F) calculated using the equation [1] may be equated withthe tension of the tether (T₀) in the equation [2] (or equation [3]). Byequating the tension of the tether with the drag force, the equation [2]may represent a scenario in which any tension of the tether is due tothe drag force on the aircraft and the weight of the tether. In theexample, an actuator of the aircraft may be positioned to provide athrust in a direction substantially perpendicular to the ground. Byproviding thrust in a substantially vertical direction, the actuator maysave power for vertical propulsion that may otherwise be used to producean additional tension on the tether. For x>0, the equation [2] generallydefines a path of the tether in which the altitude (h) of the tetherincreases as the horizontal position (x) of the tether increases. A rateof increase of the altitude with respect to the horizontal position mayincrease as the horizontal position increases. It should be noted that apath of the tether will have a finite length limited by the length ofthe tether, whereas the equation [2] and the equation [3] define analtitude for all positive and negative values of horizontal position.

The relationship between the altitude and the horizontal position of thetether may also be expressed using the equation [3]. A constant “ε” mayrepresent Euler's number or a base of a natural logarithm (approximately2.71828). The equation [3] may represent a function relating thehorizontal position of the tether and the altitude of the tether that isequivalent to the function represented by the equation [2]. Otherequations or functions that define the relationship between thehorizontal position of the tether and the altitude of the tetherequivalent to the equation [2] and the equation [3] may exist.

The first example catenary path 402 and the second example catenary path404 may be catenary paths representing relationships between ahorizontal position of the tether and an altitude of the tether. Thefirst example catenary path 402 and the second example catenary path 404may be determined based on varying parameters of the equation [1] andthe equation [2] (or the equation [3]). The horizontal position of thetether may be represented on an x-axis and the altitude of the tethermay be represented on an h-axis. The first example catenary path 402 orthe second example catenary path 404 may be calculated by using theequation [1] to calculate a drag force of the aircraft due to anapparent wind. Next, the equation [2] or the equation [3] may be used tocalculate the first example catenary path 402 or the second examplecatenary path 404 by using the drag force determined with the equation[1].

As depicted in FIG. 4A, the first example catenary path 402 and thesecond example catenary path 404 may have an altitude of zero (h=0) at ahorizontal position defined as x=0. A designation of an origin for atwo-dimensional space defined by horizontal position and altitude may bearbitrary. For example, x=0 may represent a horizontal position of theground station, or x=0 may represent a horizontal position at which aminimum altitude of the first example catenary path 402 or the secondexample catenary path 404 occurs. If x=0 represents the horizontalposition of the ground station and h=0 represents an altitude at whichthe tether couples to the ground station, the first example catenarypath 402 and the second example catenary path 404 may both representtether paths in which a minimum tether altitude occurs at the groundstation (x=h=0). By further example, a maximum tether altitude for thefirst example catenary path 402 and the second example catenary path 404may occur at an end of the tether coupled to the aircraft.

At least one parameter of the equation [2] (or the equation [3]) used todetermine the first example catenary path 402 may differ from aparameter of the equation [2] (or the equation [3]) used to determinethe second example catenary path 404. For example, the tension of thetether (T₀) represented by the first example catenary path 402 may beless than the tension of the tether (T₀) represented by the secondexample catenary path 404, while the weight per length of the tether (μ)represented by the first example catenary path 402 and the secondexample catenary path 404 may be equal. A difference in tether tensionrepresented by the first example catenary path 402 and the secondexample catenary path 404 may be caused by a difference in the densityof air (ρ), the drag coefficient (C_(d)), the reference area (A), or thespeed of the apparent wind (v), as depicted in the equation [1].Alternatively, the tension of the tether (T₀) represented by the firstexample catenary path 402 may be equal to the tension of the tether (T₀)represented by the second example catenary path 404, while the weightper length of the tether (μ) represented by the first example catenarypath 402 may be greater than the weight per length of the tether (μ)represented by the second example catenary path 404. By further example,a quantity (T₀/μ) corresponding to the second example catenary path 404may be twice that of a quantity (T₀/μ) corresponding to the firstexample catenary path 402. The variation in (T₀/μ) for the first examplecatenary path 402 and the second example catenary path 404 may be basedon varying weights per length of tethers (μ), or based on a differencein tether tensions T₀, which may be caused by differing drag forces (F).

An equation [4] may resemble equation [2], but may further include anh-axis parameter (a) and an x-axis parameter (b):

$\begin{matrix}{h = {\frac{T_{0}}{\mu}\lbrack {{\cosh ( \frac{\mu ( {x - b} )}{T_{0}} )} - ( {1 - {a\frac{\mu}{T_{0}}}} )} \rbrack}} & \lbrack 4\rbrack\end{matrix}$

The h-axis parameter (a) may be determined so that a minimum altitude ofa catenary path may occur at a specific altitude above (or below) apoint defined as h=0. For example, if a=5, then a minimum tetheraltitude of a catenary path defined by the equation [4] may occur ath=5. By further example, if b=7, then a minimum tether altitude of acatenary path defined by the equation [4] may occur at x=7. An equation[5] may also include an h-axis parameter (a) and an x-axis parameter (b)that affect a catenary path similarly to the h-axis parameter (a) andthe x-axis parameter (b) of the equation [4]:

$\begin{matrix}{h = {\frac{T_{0}}{\mu}\lfloor {( \frac{( {^{(\frac{\mu \; {({x - b})}}{T_{0}})} + ^{- {(\frac{\mu \; {({x - b})}}{T_{0}})}}} )}{2} ) - ( {1 - {a\frac{\mu}{T_{0}}}} )} \rfloor}} & \lbrack 5\rbrack\end{matrix}$

FIG. 4B depicts a third example catenary path 406, and a fourth examplecatenary path 408. The third example catenary path 406 may be defined bythe equation [4] with an h-axis parameter (a) of a₁ and an x-axisparameter (b) of b₁. As depicted in FIG. 4B, the third example catenarypath 406 may be defined by substituting (b₁) for (b) and (a₁) for (a) inthe equation [4], resulting in

$\begin{matrix}{h = {\frac{T_{0}}{\mu}\lbrack {{\cosh ( {\frac{\mu}{T_{0}}( {x - b_{1}} )} )} - ( {1 - {a_{1}\frac{\mu}{T_{0}}}} )} \rbrack}} & \lbrack 6\rbrack\end{matrix}$

which may define an altitude of h=a₁ at a horizontal position x=b₁. Apoint (h=a₁, x=b₁) on the third example catenary path 406 may correspondto a minimum tether altitude for the third example catenary path 406. Inthis case, h=0 may represent the ground and h(x=0) may represent analtitude at which the tether couples to the ground station which, basedon the equation [6], may be

$\begin{matrix}{{h( {x = 0} )} = {{\frac{T_{0}}{\mu}\lbrack {{\cosh ( {\frac{\mu}{T_{0}}( {- b_{1}} )} )} - ( {1 - {a_{1}\frac{\mu}{T_{0}}}} )} \rbrack}.}} & \lbrack 7\rbrack\end{matrix}$

As depicted in FIG. 4B, the fourth example catenary path 408 may have analtitude of h=a₂ at a horizontal position x=b₂, which may be a minimumtether altitude for the fourth example catenary path 408. The fourthexample catenary path 408 may be defined by the equation [4] and anh-axis parameter (a) of (a₂) and an x-axis parameter (b) of (b₂). Inthis case h=0 may represent the ground and h(x=0) may represent analtitude at which the tether couples to the ground station which, basedon the equation [4], may be

$\begin{matrix}{{h( {x = 0} )} = {\frac{T_{0}}{\mu}\lbrack {{\cosh ( {\frac{\mu}{T_{0}}( {- b_{2}} )} )} - ( {1 - {a_{2}\frac{\mu}{T_{0}}}} )} \rbrack}} & \lbrack 8\rbrack\end{matrix}$

The third example catenary path 406 and the fourth example catenary path408 may be portions of a same curve translated to accommodate differingdefinitions of the origin of the two-dimensional space of altitude andhorizontal position. That is, the third example catenary path 406 andthe fourth example catenary path 408 may be defined by an equal tensionof the tether (T₀) and weight per length of the tether (μ), but differonly in the h-axis parameters and x-axis parameters that define thethird catenary path 406 and the fourth example catenary path 408.

The catenary paths illustrated in FIG. 4 are examples only, and thecatenary paths may vary based on varying parameters of Equations[1]-[8].

Causing the aircraft to travel a catenary trajectory may allow theactuator of the aircraft to provide thrust in a substantially verticaldirection, allowing the drag force of the apparent wind to provide aforce to move the aircraft in a horizontal direction. To maintain ahover orientation in which the actuator is configured to provide asubstantially vertical thrust, a control surface of the aircraft may beused to adjust an orientation of the aircraft to the hover orientation,allowing the actuator to expend energy to produce a substantiallyvertical thrust.

FIG. 5A depicts an example roll axis 502 of an aircraft 530. In oneembodiment, the aircraft 530 may include actuators positioned to apply atorque thrust to the aircraft 530 about the roll axis 502 of theaircraft 530, causing the aircraft 530 to rotate about the roll axis502. To land and couple the aircraft 530 onto the ground station it maybe useful for the aircraft 530 to assume a particular roll angle withrespect to a reference roll angle. During forward flight, rolladjustments of aircraft 530 may be made by changing a position of flapson the main wing of the aircraft 530. It should be noted that thedefinition of the roll axis 502 is arbitrary and the roll axis 502 mayconstitute a different axis in another embodiment.

FIG. 5B depicts an example pitch axis 504 of the aircraft 530. Theaircraft 530 may include actuators 534A-D positioned to apply a torquethrust about the pitch axis 504 of the aircraft 530. To pitch theaircraft 530 in a negative direction, the actuators 534A and 534C mayprovide thrust while the actuators 534B and 534D are idle.Alternatively, the aircraft 530 may be pitched in a positive directionby causing the actuators 534B and 534D to provide thrust and causing theactuators 534A and 534C to be idle. Using the actuators 534A-D toprovide pitch control for the aircraft 530 may be useful during hoverflight, during which the tail wing of the aircraft 530 may not beconfigured to provide a torque about the pitch axis 504 of the aircraft530. It should be noted that definitions of positive and negative pitchand the pitch axis 504 are arbitrary and not meant to be limiting. Thepitch axis 504 may constitute a different axis in another embodiment.

FIG. 5C depicts an example yaw axis 506 of the aircraft 530. Theaircraft 530 may include the actuators 534A-D positioned to apply atorque thrust about the yaw axis 506 of the aircraft 530. To yaw theaircraft 530 in a negative direction, the actuators 534C and 534D mayprovide thrust while the actuators 534A and 534B are idle.Alternatively, the aircraft 530 may be yawed in a positive direction bycausing the actuators 534A and 534B to provide thrust and causing theactuators 534C and 534D to be idle. Using the actuators 534A-D toprovide yaw control may be useful during hover flight during which thevertical stabilizer of the aircraft 530 may not be configured to providea torque about the yaw axis 506 of the aircraft 530. It should be notedthat definitions of positive and negative yaw and the yaw axis 506 arearbitrary and not meant to be limiting. The yaw axis 506 may constitutea different axis in another embodiment.

FIG. 6A depicts examples of a pitch axis 602 of an aircraft 630, a tailwing 636, and an apparent wind 650. At times, it may be useful to changea pitch angle of the aircraft 630. To change the pitch angle of theaircraft 630 and conserve power otherwise consumed by an actuator of theaircraft 630, the tail wing 636 may be configured to orient a surface ofthe tail wing 636 to face the apparent wind 650 so that the apparentwind 650 applies a drag force to the tail wing 636. The drag force mayresult in a torque moment that causes the aircraft 630 to rotate withrespect to the pitch axis 602 in a direction indicated in FIG. 6A.

FIG. 6B depicts examples of a pitch axis 602 of an aircraft 630, a tailwing 636, and an apparent wind 650. At times, it may be useful to changea pitch angle of the aircraft 630. To change the pitch angle of theaircraft 630 and conserve power otherwise consumed by an actuator of theaircraft 630, the tail wing 636 may be configured to orient a surface ofthe tail wing 636 to face substantially perpendicular to the apparentwind 650 so that the apparent wind 650 applies a lift force 652 to thetail wing 636. The lift force 652 may result in a torque moment thatcauses the aircraft 630 to rotate with respect to the pitch axis 602 ina direction indicated in FIG. 6B.

FIG. 7 is a block diagram of an example method 700 for determining atrajectory and an orientation of the aircraft that causes a tether tounfurl along a catenary path above ground, in accordance with at leastsome embodiments described herein. Method 700 shown in FIG. 7 presentsan embodiment of a method that, for example, could be used with acomputing device. Functions of the method 700 may be fully performed bya processor of a computing device, by a computing device, or may bedistributed across multiple processors or multiple computing devicesand/or a server. In some examples, the computing device may receiveinformation from sensors of the computing device, or where the computingdevice is a server the information can be received from another devicethat collects the information.

Method 700 may include one or more operations, functions, or actions asillustrated by one or more blocks of 702-710. Although the blocks areillustrated in a sequential order, these blocks may in some instances beperformed in parallel, and/or in a different order than those describedherein. Also, the various blocks may be combined into fewer blocks,divided into additional blocks, and/or removed based on the desiredimplementation.

In addition, for the method 700 and other processes and methodsdisclosed herein, the flowchart shows functionality and operation of onepossible implementation of present embodiments. In this regard, eachblock may represent a module, a segment, or a portion of program code,which includes one or more instructions executable by a processor forimplementing specific logical functions or steps in the process. Theprogram code may be stored on any type of computer readable medium, forexample, such as a storage device including a disk or hard drive. Thecomputer readable medium may include a non-transitory computer readablemedium, for example, such as computer-readable media that stores datafor short periods of time like register memory, processor cache, andRandom Access Memory (RAM). The computer readable medium may alsoinclude non-transitory media, such as secondary or persistent long termstorage, like read only memory (ROM), optical or magnetic disks, orcompact-disc read only memory (CD-ROM), for example. The computerreadable media may also be any other volatile or non-volatile storagesystems. The computer readable medium may be considered a computerreadable storage medium, a tangible storage device, or other article ofmanufacture, for example.

In addition, for the method 700 and other processes and methodsdisclosed herein, each block in FIG. 7 may represent circuitry that iswired to perform the specific logical functions in the process.

At block 702, the method 700 includes determining a drag force of anapparent wind on an aircraft coupled to a ground station via a tether. Aprocessor may determine the drag force using a drag force equation, suchas equation [1]. More specifically, the processor may calculate the dragforce based on a proportionality between the drag force and a density ofair, between the drag force and a reference area of the aircraft,between the drag force and a drag coefficient, or between the drag forceand a square of the speed of the apparent wind. The drag coefficient mayindicate a resistance of the aircraft to air moving against a surface ofthe aircraft and may be dependent on a shape of the aircraft or thespeed of the apparent wind.

At block 704, the method 700 includes, based on the drag force and aweight of the tether, determining a trajectory of the aircraft to apoint downwind of the ground station such that the aircraft travellingthe trajectory causes the tether to unfurl along a catenary path aboveground. The catenary path may represent a shape of the tether caused bygravity acting on the tether while the tether is supported at a firstend by the ground station and supported at a second end by the aircraft.The processor may determine the trajectory by determining an azimuthangle for the trajectory that is parallel to a direction of the apparentwind. The processor may further determine a series of altitudes thatcorrespond to a series of horizontal positions of the tether along theazimuth angle.

The processor may determine the trajectory based on the drag forceequation and a catenary equation, such as equations [1]-[8] such thatthe aircraft travelling the trajectory in the apparent wind causes atension of the tether to have a horizontal component substantially equalto the drag force of the apparent wind. In this way, the trajectory maybe optimized so that the tether is maintained above a minimum altitude,the apparent wind pushes the aircraft in a horizontal direction, and theactuator of the aircraft provides a vertical thrust substantiallyperpendicular to the ground. The trajectory determined by the processormay also cause a tension of the tether to have a vertical componentequal to a weight of a portion of the tether. That is, the aircrafttravelling the trajectory may restrain the tether from touching theground or from dropping below a certain altitude, but may not requirethrust to be provided by an actuator to place additional tension on thetether.

The aircraft travelling the trajectory may also cause a first portion ofthe tether to occupy a position on the catenary path previously occupiedby a second portion of the tether. As the aircraft travels thetrajectory and increases a distance of the aircraft from the groundstation, the tether may be reeled out by the ground station toaccommodate the increased distance of the aircraft from the groundstation. An overall shape of the tether suspended by the ground stationand the aircraft may remain unchanged as the tether is reeled out,except that an additional section of the catenary path adjacent to theaircraft may be added to a previous path of the tether. In this way,once a position on the catenary path has been occupied by a portion ofthe tether, the position may continue to be occupied by other portionsof the tether as the tether is reeled out.

The method 700 may also include the processor receiving datarepresenting a horizontal position of the ground station, an altitude ofthe ground station, and a length of the tether. The processor may thendetermine the catenary path based on the length of the tether and atension of the tether so that the aircraft travelling the catenary pathcauses a tension of the tether to be about equal to the drag force ofthe apparent wind on the aircraft. The tension may occur at the minimumaltitude of the catenary path. By minimizing the tension of the tethercaused by the actuator of the aircraft, an energy dissipated by theactuator may be minimized. The processor may then determine parametersof the catenary path that cause the catenary path to include a pointdefined by the horizontal position of the ground station and thealtitude of the ground station. The processor may also determine theparameters so that a minimum altitude of the catenary path occurs withina range of horizontal position bounded by the horizontal position of theground station and the point downwind of the ground station. Theprocessor may receive data representing a minimum tether altitude anddetermine the parameters so that a minimum altitude of the catenary pathis about equal to the minimum tether altitude.

The catenary path may be determined by the processor based on theequation [4] (or equation [5]). (T₀) may represent a tension of thetether at a lowest point of the catenary path, or a horizontal componentof the tension at any point on the tether. To reduce the energy consumedby the actuator, the tension T₀ may be about equal to the drag force ofthe apparent wind defined by the equation [1]. (ρ) may represent adensity of air, (C_(d)) may represent the drag coefficient of theaircraft, (A) may represent the reference area of the aircraft, and (v)may represent the speed of the apparent wind. In the catenary equation,(μ) may represent the weight per length of the tether, (a) may representa vertical adjustment parameter, (b) may represent a horizontaladjustment parameter, (h) may represent altitude, and (x) may representhorizontal position.

For example, the horizontal position of the ground station and thealtitude of the ground station may be x=0 and h=5, respectively. Forpurposes of illustration, a quantity (T₀/μ) may be equal to 1. In thiscase, the catenary equation may take a simplified form, h=cosh(x−b)−(1−a). The processor may then determine the parameters (a) and(b) such that the altitude of the tether at a horizontal positionrepresented by x=0 is h=5. The processor may first determine (a) suchthat a minimum altitude of the catenary path corresponds to the minimumtether altitude. For example, to yield a catenary path in which theminimum altitude of the path is h=1, the processor may determine (a) tobe equal to 1, based on a minimum value of cos h(x−b) being equal to 1.The catenary equation may then be expressed as h=cos h(x−b). Next, theprocessor may determine (b) such that an altitude of the catenary pathat x=0 is h=5, by solving an equation 5=cos h(0−b). There may exist twosuch values of (b) that solve the equation, b≈2.29243 and b≈−2.29243.The processor may determine that determining (b) to be equal to 2.29243will cause the minimum altitude of the catenary path to occur at aposition between the ground station and the point downwind of the groundstation (i.e. the minimum altitude may occur on a positive-x side of thex-axis). In this example, (b) may be determined to be 2.29243 by theprocessor. By further example, referring to FIG. 4, the third examplecatenary path 406 may depict a catenary path corresponding to parametersb=1 and a=2, while the fourth example catenary path 408 may depict acatenary path corresponding to parameters b=3 and a=4. (Note that inthis example, the x-axis and the h-axis may not share a common scale.)Accordingly, the altitude of the third example catenary path 406 at x=0may be h≈2.543 and the altitude of the fourth example catenary path 408at x=0 may be h≈13.068.

The processor may also determine a horizontal position and an altitudecorresponding to an endpoint of the trajectory based on the length ofthe tether, the weight of the tether, and the drag force. Once thecatenary path is determined, an arc length (s) of the catenary path froma horizontal position x₁ to a horizontal position x₂ can be determinedusing an equation [9]:

$\begin{matrix}{{S = {\int_{x_{1}}^{x_{2}}\sqrt{1 + {( \frac{h}{x} )^{2}{x}}}}}\ } & \lbrack 9\rbrack\end{matrix}$

where (h) is the altitude of the catenary path defined by the equation[4]. If a total length of the tether is known, (s) in the equation [9]can be set equal to the total tether length, and a horizontal distancebetween the ground station at x₁ and the point downwind of the groundstation at x₂ can be determined. For purposes of illustration, theground station may have a horizontal position x=0=x₁, the tether mayhave a length of 50 and (a) may equal 2 and (b) may equal 1, yieldingequation [10]:

h(x)=cos h(x−1)−(1−2)  [10]

In this case, the endpoint of the trajectory would be determined usingan equation

50=∫₀ ^(x) ² √{square root over (1+sin h ²(x−1))}dx  [11]

A solution to the equation [11] may be x₂≈5.582. An altitude (h) of theendpoint of the catenary path may be determined by the processor usingthe equation [4] and the horizontal position of the endpoint. In thecase of x₂=5.582, h may be approximately 49.86.

At block 706, the method 700 includes determining an orientation of theaircraft to travel the trajectory in the apparent wind so that anactuator of the aircraft is configured to provide a vertical thrust in adirection substantially perpendicular to the ground. The processor mayfirst receive data representing a direction in which the actuator isconfigured to provide thrust relative to an axis of the aircraft. Next,the processor may determine an angle of rotation of the aircraftrelative to the axis of the aircraft such that at the angle of rotation,the actuator is configured to provide the vertical thrust in a directionsubstantially perpendicular to the ground. In other words, the processormay determine an orientation of the aircraft based on a relativeorientation of the actuator with respect to the aircraft, such that theactuator is configured to provide a substantially downward thrustperpendicular to the ground. Limiting the thrust of the actuator to bein the vertical direction may allow the aircraft to rely on the force ofthe apparent wind to travel in the horizontal direction.

At block 708, the method 700 includes determining a vertical thrust forthe aircraft at the orientation to travel the trajectory in the apparentwind based on the trajectory and a weight of the aircraft. The processormay also receive data representing a weight, a position, and a verticalvelocity of the aircraft, and a weight of a portion of the tethersupported by the aircraft. With the data, the processor may determine agravitational force acting on the aircraft based on the weight of theaircraft and the weight of the portion of the tether supported by theaircraft. The processor may determine the weight of the portion of thetether supported by the aircraft based on a weight per length of thetether and a length of the portion of the tether. The processor may nextdetermine a vertical acceleration of the aircraft based on the positionand the vertical velocity of the aircraft, wherein the aircraftachieving the vertical acceleration and the drag force pushing theaircraft horizontally cause the aircraft to follow the trajectory.Finally, the processor may determine the vertical thrust based on aforce to counteract the downward force and achieve the verticalacceleration.

At block 710, the method 700 includes providing instructions to causethe actuator of the aircraft to provide the vertical thrust to move theaircraft along the trajectory. The processor may provide theinstructions to the actuator or a control system of the aircraft thatcontrols the actuator.

The processor may further receive data indicating an initial orientationof the aircraft, and a speed and a direction of the apparent wind. Theprocessor may use the data to determine a position of the tail wingrelative to the direction of the apparent wind configured to cause theapparent wind to produce a rotational force about a pitch axis of theaircraft. The rotational force may be configured to rotate the aircraftfrom the initial orientation to a hover orientation. As shown anddescribed in FIGS. 6A and 6B, the tail wing 636 of the aircraft may beconfigured to provide pitch control while the aircraft is in a hoverorientation. The tail wing 636 may provide pitch control in a firstdirection by orienting the tail wing so that the apparent wind producesa drag force against the tail wing. The drag force may create a pitchmoment in a first direction about the pitch axis of the aircraft, asshown in FIG. 6A. The tail wing may provide pitch control in a seconddirection by orienting the tail wing so that the apparent wind producesa lift force against the tail wing. The lift force may create a pitchmoment in a second direction about the pitch axis of the aircraft, asshown in FIG. 6B. Lastly, the processor may provide instructions to thecontrol system of the aircraft (or the ground station) to move the tailwing to provide the rotational force to rotate the aircraft to the hoverorientation.

The tail wing may be configured to produce the lift force based on theapparent wind achieving a threshold speed, such as 15 meters per second.The processor may provide the instructions to move the tail wing toprovide the rotational force based on receiving a notification from asensor of the aircraft that the speed of the apparent wind is greaterthan or equal to the threshold speed. Unless the apparent wind has aspeed greater than the threshold speed, the tail wing may not beconfigured to provide a lift force configured for pitch control of theaircraft while the aircraft is in the hover orientation. Deploying theaircraft along the catenary path and using the tail wing for pitchcontrol may increase a margin between a nominal actuator output and amaximum actuator output, thereby increasing an ability of the aircraftto respond to disturbances (e.g. wind gusts) that cause deviations fromthe catenary path or a particular attitude of the aircraft.

It should be understood that arrangements described herein are forpurposes of example only. As such, those skilled in the art willappreciate that other arrangements and other elements (e.g. machines,interfaces, functions, orders, and groupings of functions, etc.) can beused instead, and some elements may be omitted altogether according tothe desired results. Further, many of the elements that are describedare functional entities that may be implemented as discrete ordistributed components or in conjunction with other components, in anysuitable combination and location, or other structural elementsdescribed as independent structures may be combined.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopebeing indicated by the following claims, along with the full scope ofequivalents to which such claims are entitled. It is also to beunderstood that the terminology used herein is for the purpose ofdescribing particular embodiments only, and is not intended to belimiting.

1. A method comprising: sensing a speed of an apparent wind on anaircraft, wherein the aircraft is coupled to a ground station via atether; based on the sensed speed of the apparent wind and Iii) a weightof the tether, determining, by one or more processors, a trajectory ofthe aircraft to a point downwind of the ground station such that theaircraft travelling the trajectory pulls the tether along a catenarypath above ground; based on the trajectory and a weight of the aircraft,determining a vertical thrust for the aircraft to travel the trajectoryin the apparent wind; and providing instructions to cause an actuator ofthe aircraft to provide the vertical thrust to move the aircraft alongthe trajectory.
 2. The method of claim 1, further comprising:determining a drag force of the apparent wind on the aircraft based on adensity of air, a reference area of the aircraft, or the sensed speed ofthe apparent wind; and determining the trajectory based on the dragforce.
 3. The method of claim 1, further comprising: determining a dragforce of the apparent wind on the aircraft based on a drag coefficient,wherein the drag coefficient indicates a resistance of a surface of theaircraft to air moving against the surface; and determining thetrajectory based on the drag force.
 4. The method of claim 1, whereinthe aircraft travelling the trajectory comprises a point common to boththe tether and the aircraft travelling the trajectory.
 5. The method ofclaim 1, wherein determining the trajectory of the aircraft comprises:determining an azimuth angle for the trajectory; and determining analtitude corresponding to a horizontal position on the ground along aline defined by the azimuth angle.
 6. The method of claim 1, whereindetermining the trajectory of the aircraft comprises: determining thetrajectory such that the aircraft traveling the trajectory causes atension of the tether to have a horizontal component substantially equalto a drag force of the apparent wind on the aircraft.
 7. The method ofclaim 1, wherein determining the trajectory of the aircraft comprises:determining the trajectory such that the aircraft traveling thetrajectory causes a tension of the tether to have a vertical componentsubstantially equal to a weight of the tether.
 8. The method of claim 1,wherein determining the trajectory of the aircraft comprises:determining the trajectory such that the aircraft traveling thetrajectory causes a first portion of the tether to occupy a position onthe catenary path previously occupied by a second portion of the tether.9. The method of claim 1, wherein determining the trajectory of theaircraft comprises: determining the catenary path based on a length ofthe tether and a tension of the tether, wherein the tension of thetether is about equal to a drag force of the apparent wind on theaircraft; and determining a parameter of the catenary path so that: thecatenary path includes a point defined by the horizontal position of theground station and the altitude of the ground station, and a minimumaltitude of the catenary path occurs within a range of horizontalposition bounded by the horizontal position of the ground station andthe point downwind of the ground station.
 10. The method of claim 9,wherein determining the trajectory of the aircraft further comprises:determining the parameter of the catenary path so that the minimumaltitude of the catenary path is about equal to a predeterminedaltitude.
 11. The method of claim 1, wherein determining the trajectoryof the aircraft comprises: determining a horizontal positioncorresponding to an endpoint of the trajectory based on a length of thetether, the weight of the tether, and a drag force of the apparent windon the aircraft.
 12. The method of claim 1, wherein determining thetrajectory of the aircraft comprises: determining an altitudecorresponding to an endpoint of the trajectory based on a length of thetether, the weight of the tether, and a drag force of the apparent windon the aircraft.
 13. The method of claim 1, further comprising:determining an orientation of the aircraft to travel the trajectory inthe apparent wind so that the actuator of the aircraft is configured toprovide the vertical thrust in a direction substantially perpendicularto the ground.
 14. The method of claim 1, wherein determining thevertical thrust for the aircraft comprises: determining the verticalthrust based on a position and a vertical velocity of the aircraft,wherein the actuator providing the vertical thrust and a drag force ofthe apparent wind on the aircraft pushing the aircraft horizontallycause the aircraft to follow the trajectory.
 15. The method of claim 1,wherein the aircraft includes a tail wing and is engaged in hoverflight, and the method further comprises: receiving data indicating aninitial orientation of the aircraft, and a speed and a direction of theapparent wind; determining a position of the tail wing relative to thedirection of the apparent wind configured to cause the apparent wind toproduce a rotational force about a pitch axis of the aircraft to rotatethe aircraft from the initial orientation to a hover orientation; andproviding instructions to move the tail wing to provide the rotationalforce to rotate the aircraft to the hover orientation.
 16. The method ofclaim 15, wherein providing instructions to move the tail wing toprovide the rotational force to rotate the aircraft to the hoverorientation comprises: providing the instructions based on receiving anotification that a speed of the apparent wind is sufficient to producethe rotational force.
 17. A non-transitory computer readable storagememory having stored therein instructions, that when executed by acomputing device that includes one or more processors, causes thecomputing device to perform functions comprising: sensing a speed of anapparent wind on an aircraft, wherein the aircraft is coupled to aground station via a tether; based on (i) the sensed speed of theapparent wind and (ii) a weight of the tether, determining a trajectoryof the aircraft to a point downwind of the ground station such that theaircraft travelling the trajectory pulls the tether along a catenarypath above ground; based on the trajectory and a weight of the aircraft,determining a vertical thrust for the aircraft to travel the trajectoryin the apparent wind; and providing instructions to cause an actuator ofthe aircraft to provide the vertical thrust to move the aircraft alongthe trajectory.
 18. The non-transitory computer readable storage memoryof claim 17, wherein the aircraft travelling the trajectory comprises apoint common to both the tether and the aircraft travelling thetrajectory.
 19. A computing device comprising: one or more processors;and memory configured to store instructions, that when executed by theone or more processors, cause the computing device to perform functionscomprising: sensing a speed of an apparent wind on an aircraft, whereinthe aircraft is coupled to a ground station via a tether; based on (i)the sensed speed of the apparent wind and (ii) a weight of the tether,determining a trajectory of the aircraft to a point downwind of theground station such that the aircraft travelling the trajectory pullsthe tether along a catenary path above ground; based on the trajectoryand a weight of the aircraft, determining a vertical thrust for theaircraft to travel the trajectory in the apparent wind; and providinginstructions to cause an actuator of the aircraft to provide thevertical thrust to move the aircraft along the trajectory.
 20. Thecomputing device of claim 19, wherein the aircraft travelling thetrajectory comprises a point common to both the tether and the aircrafttravelling the trajectory.